Purgatory

Purgatory is where people think other people go if they are not good enough for Heaven or bad enough for Hell.

Roman Catholics

Roman Catholics believe that those guilty of minor sins need purification before they become fit to be with God in Heaven and this purification involves severe torture for a finite period. The Bible does not mention Purgatory but Roman Catholics interpret some passages as referring to it.

If the work which any man has built on the foundation [which is Christ] survives, he will receive a reward. If any man's work is burned up, he will suffer loss, though he himself will be saved, but only as through fire" (1 Cor 3: 14-15). [1]

There is a similar problem as with the Problem of Hell as to why a loving God cannot purify sinners without inflicting extreme suffering.
Roman Catholics pray for the souls of departed loved-ones, they also pay priests to say mass and ease the souls of deceased loved ones through Purgatory.

Protestants

Most Protestants don't believe in Purgatory and some have no clear idea what happens to those guilty of small sins. Some extreme Protestants believe that every sin however small merits Hell and they live in perpetual fear that they will end up in Hell over something or other while their church uses the fear to control them. Other Protestants believe that once they've accepted Jesus they're saved and seem to believe that they're assured of Heaven but they have to live good lives as well. It gets confusing and it's not clear what they think happens over sins committed after they're saved.